Project 1: Match-Up

In ice-hockey, an important aspect of the game is choosing match-ups against the opposing team. One team (the home team) gets to choose whom to play after seeing which opponents have been selected. We'll be simulating a variant of ice-hockey in which each team has three ``lines'' each of five players. The net skill level of the two teams will be equally matched, and the aim of the game is to make the most of the match-up advantage to score more often.

Each group manages a team of fifteen players. Each player has an integer skill level between 1 and 11, and the total skill level must add to 90. The average skill level of all players is therefore 6. Each group selects in advance the skill level of all 15 players before the games start. We'll run games in pairs, with the same distribution of skill levels in both games, so that each team gets a chance to be the home team.

At the beginning of each game, each group divides their players into three lines, knowing the skill distribution of the opponent, but not knowing the opponent's line composition. This division into lines does not change once the game commences. Each group knows whether they are the home team or the away team for a game when they divide their players into lines.

The game proceeds as follows. The away team puts a line on the ice, and the home team selects a line to match up against the away team, knowing the skill levels of the opponent's players. Further, the home team can permute the order of the players to achieve any specific player match-up desired. For example, suppose the away team puts out a line containing skill levels (4,5,6,7,8). The home team might have a line with exactly the same skill levels, but order them (7,8,4,5,6). This process happens twice more, for the second and third lines respectively.

The score generated by a particular pair of lines is computed as follows: Suppose that at a position p (from 1 to 5), team 1's player has skill s₁ and team 2's player has skill s₂. Team 1 wins that position when $s_1-s_2 \geq 3$. Team 2 wins that position when $s_2-s_1 \geq 3$. Neither team wins the position when |s₁ - s₂ | < 3. In the example above, the home team would win positions 1 and 2, while the remaining three positions would have no winner. The score for each team in a game is the number of positions won, aggregated over the three lines.

The net score over the two games is used as the final score between the two groups. We'll run a bunch of game-pairs between all pairs of groups in a tournament at the end of the project. Note that groups can choose different sets of 15 players for each pair of games, so that the results of games against the same opponent may vary. In the tournament, players will be able to keep a history of outcomes of their games to help them select a team distribution for subsequent games.

Some things to think about:

• Is it possible for three lines L₁, L₂, and L₃ to exist such that L₁ always beats L₂, L₂ always beats L₃ and L₃ always beats L₁?
• Are the ``good'' lines different for home and away teams?
• How do you choose a skill distribution that allows you to choose ``good'' home-team and away-team lines?

This game was motivated by my experience in high school team chess tournaments. While teams were expected to rank the five players in order of playing strength to determine match-ups, a team could conceivably get an (unfair?) advantage against a team following the rules by putting their worst player first and improving the pairwise skill advantage on the other boards.